282 ANALYTICAL MECHANICS 3. A circular table is perfectly free4 to rotate about a vortical a: through its center. Show that if a man walks completely around the ed of the table the latter turns through an angle of ^ t) - - 2 TT, where .and M are the masses of the table and of the man, respectively. 4. A circular plate is rotating about its axis, which is vertical, with ; angular velocity co and is moving on a smooth horizontal piano with a line velocity ?;. Find the angular velocity it; will have if one of the elemoi. of its lateral surface is suddenly fixed, and determine the impulse giv to the axis of rotation. Discuss the results for special positions of t fixed axis. 5. A uniform rod strikes at one end against an obstacle while falli transversely. Show that the impulse which the obstacle receives \v be one-half that which it would have received if the other (Hid of t rod had struck an obstacle simultaneously with the first. 6. A particle is projected into a tube which is bent, to form a circle ai is lying on a smooth horizontal table. If the inner surface of the tube perfectly smooth, show that the center of mans of the two moves in t direction of projection of the particle with a velocity of . --*>, wh the particle and the center of the. tube describe, circles about it with ; .angular velocity -, where M is the mass and a the radius of the tul ft while m is the mass and v the velocity of projection of the particle. 7. Find the direction and point of application which an impulse mi: have in order to make a sphere rotate about a tangent. 8. A uniform rod which is rotating on a smooth horizontal piano abo a pivot through its middle point breaks into two equal parts. Determi the subsequent motion. 9. A uniform rod rotates on a smooth horizontal plane about a piv< What will be the motion when the pivot breaks? 10. A uniform rod falls from a position where its lower end is in e<j tact with a rough horizontal plane with which it makes an angle a. Sh< that when it becomes horizontal its angular velocity is y' ^ ĽKl!l-~, wlu I/ I is the length of the rod. 11. Show that in problem (10) the angular velocity will be the sai when the horizontal plane is smooth. 12. A uniform rod which lies on a smooth horizontal plane is stru at one end, transversely. Show that the energy imparted equals | of t energy which would have been given to the bar by the same blow if t other end of the bar were fixed.